Complete DiGeorge anomaly (cDGA) is a congenital disorder characterized by athymia. Without successful treatment, children remain immunodeficient and usually die by age 2 years. In complete DiGeorge subjects, thymus transplantation with and without immunosuppression has resulted in diverse T cell development and good T cell function. The purpose of this Phase I/II study is to continue thymus transplantation safety and efficacy research for the treatment of complete DiGeorge anomaly. Until thymus transplantation is FDA approved as standard care for DiGeorge anomaly, research study participation is the only means by which a patient may have access to this potentially life-saving procedure.